Nagell-Lutz theoremNagellLutzTheorem2003-08-18 10:35:402003-08-18 10:35:40TheoremMazur's theorem on torsion of elliptic curvesNagell-Lutz theoremtorsionelliptic curveMazur's theorem% this is the default PlanetMath preamble. as your knowledge
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% Some sets
\newcommand{\Nats}{\mathbb{N}}
\newcommand{\Ints}{\mathbb{Z}}
\newcommand{\Reals}{\mathbb{R}}
\newcommand{\Complex}{\mathbb{C}}
\newcommand{\Rats}{\mathbb{Q}}The following theorem, proved independently by E. Lutz and T.
Nagell, gives a very efficient method to compute the torsion
subgroup of an elliptic curve defined over $\Rats$.
\begin{thm}[Nagell-Lutz Theorem]
Let $E/\Rats$ be an elliptic curve with Weierstrass equation:
$$y^2=x^3+Ax+B,\quad A,B\in \Ints$$
Then for all non-zero torsion points $P$ we have:
\begin{enumerate}
\item The coordinates of $P$ are in $\Ints$, i.e. $$x(P),y(P)\in
\Ints$$
\item If $P$ is of order greater than $2$, then $$y(P)^2\quad
divides\quad 4A^3+27B^2 $$
\item If $P$ is of order $2$ then $$y(P)=0\quad and\quad
x(P)^3+Ax(P)+B=0$$
\end{enumerate}
\end{thm}
\begin{thebibliography}{9}
\bibitem{lutz} E. Lutz, {\em Sur l'equation $y^2=x^3-Ax-B$ dans
les corps p-adic}, J. Reine Angew. Math. 177 (1937), 431-466.
\bibitem{nagell} T. Nagell, {\em Solution de quelque problemes
dans la theorie arithmetique des cubiques planes du premier
genre}, Wid. Akad. Skrifter Oslo I, 1935, Nr. 1.
\bibitem{milne} James Milne, {\em Elliptic Curves}, \PMlinkexternal{online course
notes}{http://www.jmilne.org/math/CourseNotes/math679.html}.
\bibitem{silverman} Joseph H. Silverman, {\em The Arithmetic of Elliptic Curves}. Springer-Verlag, New York, 1986.
\end{thebibliography}